Given, a random sample from $N(0,1)$. Investigate the behavior of $T =(X_1 + X_2)/3$ in terms of unbiasedness, consistency, efficiency and sufficiency.
What I don't understand is, there is no unknown parameter here. Then what is the estimator for?
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1$\begingroup$ You are right - have you asked the person who gave you the question? $\endgroup$– Christoph HanckDec 14, 2015 at 12:08
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$\begingroup$ No. I was working out previous year question papers. I suppose it's N(theta,1). Is sufficiency proved using the one to one property (T a one one function of sum of Xi) ? $\endgroup$– HarryDec 14, 2015 at 16:11
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1$\begingroup$ The properties of a statistical estimator do not depend on the state of mind of its user. Whether or not you happen to know the parameter, you can still investigate how well the estimator works to identify that parameter. $\endgroup$– whuber ♦Dec 16, 2016 at 19:06
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1 Answer
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This is the usual situation when using simulation to investigate properties of estimators. You have designed the simulation, so you know the values of the "unknown" parameters. Still this might be a useful exercise!
As said in a comment by user whuber:
The properties of a statistical estimator do not depend on the state of mind of its user. Whether or not you happen to know the parameter, you can still investigate how well the estimator works to identify that parameter.
answered Jul 31 at 15:03